In 2007 Maier
set down the 192 substitutions of variables which preserve the form of Heun's
equation. The table 2 of his paper [1]}
displays the transformations of
each parameter and variable of the equation.
These transformations follow from the composition of 8 homotopic
substitutions of
the dependent variable \(H(x)\)  (which keep unaltered the
independent variable
\(x\) ) with
24 linear fractional substitutions
\(x\mapsto y(x)=(Ax+B)/(Cx+D)\)  (in general these demand a transformation of the independent variable as well).
We rewrite some of these substitutions in a form convenient
for applications.
